1

The study of ordered geometry is restricted to the primitive notions of points and the relation of betweenness (with sensible axioms governing the behavior of betweenness).

A convex metric space such that, for any two distinct points $x,y \in X$, there exists a third distinct point $z$ such that: $d(x,z) + d(z,y) = d(x,y)$, i.e. the triangle inequality is an equality.

Most importantly, we say that the point $z$ is between $x$ and $y$.

Question: This strongly suggests that the notion of convex metric space was introduced in order to study ordered geometry within the framework of metric geometry.

But are convex metric spaces the only "reasonable" way to study ordered geometry within the context of metric geometry and/or metric spaces?

Or are their other reasonable ways to define the notion of betweenness in metric spaces that lead to genuine geometric insights?

Note: The most basic example of metric geometry, Euclidean geometry on $\mathbb{R}^2$, does have its ordered geometry notion of betweenness defined in terms of the convex metric space definition.

When I say "reasonable", I mean analogously to how (essentially) the only "reasonable" way to define congruence within the framework of metric geometry is to define: $$ x_1x_2 \equiv x_3 x_4 \iff d(x_1,x_2) = d(x_3,x_4) \,.$$

Community
  • 1
Chill2Macht
  • 20,920
  • Millman, Parker - Geometry: A Metric Approach with Models approaches metric geometry via incidence geometry, rather than order geometry, so this might not be relevant. But anyway, thinking in terms of incidence geometry, where the primitive notions are points and lines, not just points, we define betweenness in terms of incidence on the lines. But (for whatever reason) the lines in a metric incidence geometry don't have to correspond to the geodesics, so it is possible to have $d(A,C) = d(A,B)+d(B,C)$ for three points which are not collinear. See problem 10 on p.52 of the 2nd edition of the – Chill2Macht Jul 03 '17 at 15:11
  • aforementioned book. If the metric space's metric is equal to its intrinsic metric https://en.wikipedia.org/wiki/Intrinsic_metric , i.e. if it is a length space, then if $d(A,C) = d(A,B) + d(B,C)$, then one has (I think), that the three points $A,B,C$ lie on the same geodesic. So using the notion of betweenness in the question probably is only "natural" when: (1) the space $X$ is a length space and (2) the incidence structure (i.e. the choice of lines) on the metric space coincides with the space's geodesics. If either fails,like in the taxicab plane,then the choice might no longer be natural. – Chill2Macht Jul 03 '17 at 15:16

0 Answers0